where C is the constant of integration to be determined from S_r (0)= 0. Next we expand the term \frac{\mathbf{D}}{e^{\mathbf{D}}-1} into its power series expansion.

Definition

\frac{z}{e^z-1} = \sum_{k=0}^{\infty }B_k \frac{z^k}{k!}

Here, B_k, k=0, 1 , 2,… are known as Bernoulli’s numbers. We know that \frac{z}{e^z-1} converges when z=\pm 2\pi ni, n=1,2,3,… or when |z|=2\pi. Next, let us examine the properties of Bernoulli’s numbers.

On the left hand side there is the product of two power series. The first is from the definition of the Bernoulli numbers, and the second is the power series for the exponential function. We recall the rule for multiplying two power series. If

where (1+B)_n is to be expanded just like a binomial expansion except that instead of taking superscripts to get powers such as B_k we take subscripts to get the various Bernoulli numbers B_k. Actually since the B_n term cancels from both sides, we get a relation involving Bernoulli numbers until B_{n-1}. So